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Question: Write the change matrix $Q$ from the coordinate system $(x_i',e_i')$ to the coordinate system $(x_i'', e_i'')$, where $e_1' = (1, 2, 1)^T$, $e_2' = (1, 2, 2)^T$, $e_3' = (1, 1, 2)^T$, and $e_1'' = (1, 0, 1)^T$, $e_2'' = (1, 1,0)^T$, $e_3'' = (2, 2, 1)^T$.

My attempt to find the coordinates of $e_i'$ with respect to $\{ e_1'',e_2'',e_3''\}$

$e_i' = ae_1''+be_2''+ce_3''$

$^t(1, 2, 1)= a ^t(1, 0,1)+ b^t(1, 1, 0)+c^t(2, 2, 1)$

$^t(1, 2, 1)= ^t(a+b+2c, b+2c, a+c)$

Which $a+b+2c=1$

$b+2c=2$, leading to $b=2-2c$

$a+c=1$, leading to $a=1-c$

So solving it gives be $a=-1,b=-2,c=2$

Thus the coordinates of $e_1'$ in $\{ e_1'',e_2'',e_3''\}$ are $(-1,-2,2)$, which this will be the first column of the matrix from $(x_i',e_i')$ to $(x_i'',e_i'')$

and so on I used

$e_2' = = a ^t(1, 0,1)+ b^t(1, 1, 0)+c^t(2, 2, 1)$

$e_3' = = a ^t(1, 0,1)+ b^t(1, 1, 0)+c^t(2, 2, 1)$

to find the rest column, which I got matrix of $Q$ being

\begin{bmatrix}-1&-1&0\\-2&-4&-3\\2&3&2\end{bmatrix}

But my colleagues got different answer... is my working wrong? I calculated everything again and again and I do not think it is calculation error but something went wrong instead on the working. Many of my colleagues got

\begin{bmatrix}1&2&3\\-2&-2&-3\\2&1&2\end{bmatrix}

My colleagues could done it wrong but I doubt so because I'm the only one got this

Was I supposed to do

$e_i'' = ae_1'+be_2'+ce_3'$

instead?? Please do give feedback

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